Mesh moving incorrectly with bones
epsilonn,
I have seen the rig and the object and it really doesn't look good. It is very common and happens to a lot of people. If you are going with joining one part which is a separate object with the rig, it is not supposed to happen. If you don't understand, imagine the front is one object and the back arm is one object and both are connected to the same thing. This provides fewer problems but is kinda tiresome. If you have done this, this shouldn't happen. But if you went with the common method of making the whole body as 1 object and using automatic weights to parent it, then this problem occurs. Luckily, it is easy to fix. I will share the link to the youtube video. It's below.
https://www.youtube.com/watch?v=l9S-jhBIEiU
If you are using a version below blender 2.8, here's the link for that version:
https://www.youtube.com/watch?v=Tl4qTgwQwYw
If it still doesn't work, then you have to sadly delete the whole model and the rig and start over again. There's no other option if weight painting doesn't work.